Baire category is the hidden engine behind several major Banach-space theorems. Its role is to convert countably many pointwise or local pieces of information into one genuine open set of uniform control.
Nowhere dense and meagre sets
Category language distinguishes smallness by topology rather than measure. A nowhere dense set may be large algebraically, but it cannot contain an open piece after closure.
Definition. A subset \(A\) of a topological space \(X\) is nowhere dense if \(\operatorname{int}\overline A=\varnothing\). It is meagre, or of first category, if it is a countable union of nowhere dense sets.
Proposition. In a metric space, \(A\) is nowhere dense iff every nonempty open ball contains a smaller nonempty open ball disjoint from \(A\).
Proof. If \(A\) is nowhere dense, no nonempty open ball is contained in \(\overline A\), so \(B(x,r)\setminus\overline A\) is nonempty and open; choose a smaller ball inside it. Conversely, if every ball contains a smaller ball disjoint from \(A\), then no ball is contained in \(\overline A\), so \(\overline A\) has empty interior.
Baire theorem
The theorem says that a complete metric space cannot be assembled from countably many topologically thin pieces. This is the point where completeness becomes a structural force.
Theorem. Every complete metric space is a Baire space: countable intersections of open dense sets are dense. Equivalently, a complete metric space is not a countable union of closed sets with empty interior.
Proof. Let \(U_n\) be open dense sets and let \(B_0\) be a nonempty open ball. Since \(U_1\) is dense and open, choose a closed ball \(\overline B_1\subset B_0\cap U_1\) with radius at most \(2^{-1}\). Inductively choose \(\overline B_n\subset B_{n-1}\cap U_n\) with radius at most \(2^{-n}\).
Choose \(x_n\) the center of \(B_n\). The nested balls and shrinking radii imply \((x_n)\) is Cauchy, hence \(x_n\to x\). Since \(\overline B_n\) is closed and contains all later centers, \(x\in\overline B_n\subset U_n\) for every \(n\), and \(x\in B_0\). Thus \(B_0\cap\bigcap_nU_n\ne\varnothing\). The closed-set form follows by applying this to the complements.
Interior principle
The proof technique is usually used through an interior conclusion. Once a complete space is covered by closed sets, at least one set must contain a genuine ball.
Corollary. If a Banach space \(X\) is covered by closed sets \(F_n\), then at least one \(F_n\) has nonempty interior.
Proof. If every \(F_n\) had empty interior, then \(X=\bigcup_nF_n\) would be meagre in itself, contradicting the Baire theorem.
Riesz lemma
Riesz lemma is the geometric witness that proper closed subspaces do not fill a normed space. It is the standard way to construct separated sequences in infinite dimension.
Lemma. Let \(M\) be a proper closed subspace of a normed space \(X\). For every \(0<\alpha<1\), there exists \(x\in X\) with \(\|x\|=1\) and \(\operatorname{dist}(x,M)>\alpha\).
Proof. Choose \(y\notin M\). Since \(M\) is closed, \(d=\operatorname{dist}(y,M)>0\). Pick \(m_0\in M\) with \(\|y-m_0\|<d/\alpha\), and set \(x=(y-m_0)/\|y-m_0\|\). For any \(m\in M\), \(\|x-m\|=\|y-m_0-\|y-m_0\|m\|/\|y-m_0\|\). The numerator is the distance from \(y\) to an element of \(M\), hence at least \(d\). Therefore \(\operatorname{dist}(x,M)\ge d/\|y-m_0\|>\alpha\).
Noncompactness of the unit ball
The unit ball looks compact in finite dimension, but Riesz lemma shows why this intuition fails in infinite-dimensional normed spaces.
Theorem. The closed unit ball of an infinite-dimensional normed space is never compact in norm.
Proof. Choose inductively finite-dimensional subspaces \(M_n=\operatorname{span}\{x_1,\ldots,x_n\}\). Since \(M_n\) is proper and closed, Riesz lemma with \(\alpha=1/2\) gives \(x_{n+1}\) of norm \(1\) with \(\operatorname{dist}(x_{n+1},M_n)>1/2\). Then \(\|x_m-x_n\|>1/2\) whenever \(m>n\). The sequence in the unit ball has no Cauchy subsequence, so the unit ball is not compact.
Baire as the source of uniform theorems
The later Banach-space theorems all use the same pattern. Pointwise or local hypotheses give a countable cover, and Baire turns one member of that cover into uniform information.
Remark. The uniform boundedness principle, open mapping theorem, bounded inverse theorem, and closed graph theorem are all Baire-category theorems in disguise. Their common form is: pointwise or algebraic information becomes uniform once completeness is present.